Trigonometric Identities

Trigonometric Identities sin(theta) = a / c csc(theta) = 1 / sin(theta) = c / a cos(theta) = b / c sec(theta) = 1 / cos(theta) = c / b tan(theta) = sin(theta) / cos(theta) = a / b cot(theta) = 1/ tan(theta) = b / a sin(-x) = -sin(x) csc(-x) = -csc(x) cos(-x) = cos(x) sec(-x) = sec(x) tan(-x) = -tan(x) cot(-x) = -cot(x) sin^2(x) + cos^2(x) = 1 tan^2(x) + 1 = sec^2(x) cot^2(x) + 1 = csc^2(x) sin(x y) = sin x cos y cos x sin y cos(x y) = cos x cosy sin x sin y tan(x y) = (tan x tan y) / (1  tan x tan y) sin(2x) = 2 sin x cos x cos(2x) = cos^2(x) - sin^2(x) = 2 cos^2(x) - 1 = 1 - 2 sin^2(x) tan(2x) = 2 tan(x) / (1 - tan^2(x)) sin^2(x) = 1/2 - 1/2 cos(2x) cos^2(x) = 1/2 + 1/2 cos(2x) sin x - sin y = 2 sin( (x - y)/2 ) cos( (x + y)/2 ) cos x - cos y = -2 sin( (x - y)/2 ) sin( (x + y)/2 ) Trig Table of Common Angles angle 0 30 45 60 90 sin^2(a) 0/4 1/4 2/4 3/4 4/4 cos^2(a) 4/4 3/4 2/4 1/4 0/4 tan^2(a) 0/4 1/3 2/2 3/1 4/0 Given Triangle abc, with angles A,B,C; a is opposite to A, b opposite B, c opposite C:a/sin(A) = b/sin(B) = c/sin(C) (Law of Sines) c^2 = a^2 + b^2 - 2ab cos(C)b^2 = a^2 + c^2 - 2ac cos(B) a^2 = b^2 + c^2 - 2bc cos(A) (Law of Cosines) (a - b)/(a + b) = tan [(A-B)/2] / tan [(A+B)/2] (Law of Tangents)
Trig Table of Common Angles angle (degrees)  0  30  45  60  90  120  135  150  180  210  225  240  270  300  315  330  360 = 0 angle (radians)  0  PI/6 PI/4 PI/3 PI/2 2/3PI 3/4PI 5/6PI PI  7/6PI  5/4PI  4/3PI  3/2PI  5/3PI  7/4PI  11/6PI  2PI = 0 sin(a) (0/4) (1/4) (2/4) (3/4) (4/4) (3/4) (2/4) (1/4) (0/4) -(1/4) -(2/4) -(3/4) -(4/4) -(3/4) -(2/4) -(1/4) (0/4) COs(a) (4/4) (3/4) (2/4) (1/4) (0/4) -(1/4) -(2/4) -(3/4) -(4/4) -(3/4) -(2/4) -(1/4) (0/4) (1/4) (2/4) (3/4) (4/4) tan(a) (0/4) (1/3) (2/2) (3/1) (4/0) -(3/1) -(2/2) -(1/3) -(0/4) (1/3) (2/2) (3/1) (4/0) -(3/1) -(2/2) -(1/3) (0/4) Those with a zero in the denominator are undefined. They are included solely to demonstrate the pattern.      Hyperbolic Trigonometric Identities Hyperbolic Definitions sinh(x) = ( e x - e -x )/2csch(x) = 1/sinh(x) = 2/( e x - e -x ) cosh(x) = ( e x + e -x )/2 sech(x) = 1/cosh(x) = 2/( e x + e -x ) tanh(x) = sinh(x)/cosh(x) = ( e x - e -x )/( e x + e -x ) coth(x) = 1/tanh(x) = ( e x + e -x)/( e x - e -x ) cosh 2(x) - sinh 2(x) = 1 tanh 2(x) + sech 2(x) = 1 coth 2(x) - csch 2(x) = 1 Inverse Hyperbolic Definitions arcsinh(z) = ln( z + (z 2 + 1) )arccosh(z) = ln( z  (z 2 - 1) ) arctanh(z) = 1/2 ln( (1+z)/(1-z) ) arccsch(z) = ln( (1+(1+z 2) )/z ) arcsech(z) = ln( (1(1-z 2) )/z ) arccoth(z) = 1/2 ln( (z+1)/(z-1) ) Relations to Trigonometric Functions sinh(z) = -i sin(iz)csch(z) = i csc(iz) cosh(z) = cos(iz) sech(z) = sec(iz) tanh(z) = -i tan(iz) coth(z) = i cot(iz) Trigonometric Graphs